Last-iterate Convergence of ADMM on Multi-affine Quadratic Equality Constrained Problem

TL;DR

This paper proves that ADMM converges for a specific class of non-convex quadratic problems with multi-affine constraints, with linear convergence under certain conditions, and validates results through robotic applications.

Contribution

It establishes convergence and linear convergence rates of ADMM for multi-affine quadratic equality constrained problems, a class with practical relevance.

Findings

ADMM converges under mild assumptions for these problems.

Linear convergence is achieved when non-convexity is bounded.

Practical validation in robotic locomotion demonstrates applicability.

Abstract

In this paper, we study a class of non-convex optimization problems known as multi-affine quadratic equality constrained problems, which appear in various applications--from generating feasible force trajectories in robotic locomotion and manipulation to training neural networks. Although these problems are generally non-convex, they exhibit convexity or related properties when all variables except one are fixed. Under mild assumptions, we prove that the alternating direction method of multipliers (ADMM) converges when applied to this class of problems. Furthermore, when the "degree" of non-convexity in the constraints remains within certain bounds, we show that ADMM achieves a linear convergence rate. We validate our theoretical results through practical examples in robotic locomotion.